That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle.

Mathematics has always been highly competitive.

>"While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions without such analysis would be of negligible or even negative value for these purposes."

Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...

No matter what happened this must be a wake up call for all of us. We’re basically sharing everything we have with these companies/AI systems. This is wildly different than a human wiretapping our private messages. Because it is systematic and automated in an astronomical scale. There is no real privacy in this new world. Law? I think “National Security” is a good enough excuse to screen anything constantly, including foreign researchers in case they are close to a breakthrough.

Isn't it safe to say that all famous unsolved math problems will get a "massive amount of AI-powered effort" pointed at them regardless?

I think he's suffering from a sort of static-universe fallacy. People aren't going to keep doing what they are doing, but secretly.

What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.

Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.

In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.

Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.

Yes, that was the allegation last night.

I work at OpenAI, though not on the team that did this, and my understanding is:

- we decided to ask our model for Millenium problem solutions because of two reasons: (a) our new model was looking incredibly good and (b) we heard rumors that some Millenium problems had been solved and were curious if our models could solve them (the goal here was not to scoop any particular individuals and we were looking at many problems beyond these)

- we did not read any private chats (but of course the model was aware of prior research literature published to the internet)

- the proof generated by our model was very different from theirs and also goes far beyond the published literature

- we made an effort to jointly announce rather than immediately scoop (I understand Tristan was unhappy with the conversations; I know zero details here and I hope more is shared today)

Edit: Here's is Sebastian's take: https://x.com/SebastienBubeck/status/2097379411691516310?s=2...

That is addressed in the article.

Did you actually read the article and the substance of the solution?

>our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)

With Lean, math has become a really well suited problem for LLMs. We will likely see large gains for many years from here, just doing more and more rlvr, like continuously, non stop. No need to train from scratch. It really doesn't speak to the general intelligence of models though. It does speak to how good these things can become when a problem space has verifiable rewards, especially when you can verify one step at a time like Lean enables.

The implication from their last couple of published articles[1][2] is that they think they’ve achieved “recursive self improvement”.

[1] https://openai.com/index/research-acceleration-view-inside-o... [2] https://openai.com/index/an-alien-mind/

> Buried under the drama is the fact that OpenAI is claiming that an internal model they’ve been training for less than two weeks is more than twice as capable in mathematics as Astra.

Is this buried under the drama or are the major OpenAI twitter accounts from the people involved in the drama desperately attempting to make this the story after everything else obviously got away from them?

Not only that, but it used 10k agents coherently over 88 hours to come up with the proof. This is a significant advance.

The singularity happening under trump? We could have had star trek, instead we're getting the combine.

Can someone explain if i understand this correctly: Are they saying that they started training this new model on August 28th and then started using it on September 1st? Does training a new model only take 3 days?

Astra was trained more than two weeks ago.

I’m of zero knowledge on model training, but how is a model accessible while performing training at the same time, especially so early in its run? I’m obviously thinking a little too narrowly in terms of how it actually works

They're also counting on more casual observers to extrapolate optimistically from successes in high profile math theorems to the company's economic value.

Yeah I'm surprised they posted a chart, you would think they would keep specifics like that hidden until they're closer to launch

Yeah it's Bel

Carefully worded; it's extremely likely to be the same large frontier model that started training again on August 28th as well, as they revealed in some of the RL message board follow-up - for several reasons, most importantly, if we assume it was start of training, only a week from start of training to producing any answer would imply several orders of magnitude increase in training speed/decrease in model size.

Brain has loops and parallel connections.

Loops and parallel connections make transformer go brrr

Or they trained a LoRA on the victims chats in order to launder their plagiarism.

Pre-IPO marketing?

Most experimental physics and other natural sciences are strongly driven by their theoretical siblings, i.e. in particle research nothing gets built without a solid theoretical foundation of what you expect to find (or where you expect existing theories to break down), the same is true in other areas, no one is doing an experiment in quantum physics before they have a solid theoretical understanding of the effects they try to see. I think AI can come up with great experiments. And if epxeriments lead to results that are unexpected AI can help with that as well.

So I'm greatly excited what AI will bring about in physics, more so than in math, because in physics it's clear that our fundamental theories are missing a big piece of the picture, and given how easily AI crunches through Millenium prize problems I think it's possible that AI will come up with a viable grand unified theory uniting quantum mechanics and gravitation, or produce new predictions in other areas. There's enough contradictory or unexplained observational data available to make a ton of progress on the theory side I think. Exciting times ahead!

> I wish it were done by different folks, ideally under some kind of public control like NASA research or the NPR model.

This is sort of what OpenAI was supposed to be. I'll never understand how it was legal for them to turn it into a for profit corporation.

The problem with physics and chemistry is that you need simulations and those are often in themselves compute hungry. So the iteration loop will be slower.

I think problem with natural sciences is that it is not so easy to verify solutions to problems - there are always countless competing explanations for the data which is also often noisy - I find AI to lack the "common sense" when working with data from physical measurements .. it somehow has no touch with reality and doesn't have a feeling of the data like a domain scientist

>> Keep in mind: natural science is different. It's not always a matter of computation.

Math is like this too. The big problems they've been solving have been identified as interesting only through lots of prior effort.

There are lots of startups creating labs that can be managed e2e by agents. That will connect reasoning to the physical world and dramatically speed up the plan, experiment, reflect loop beyond what humans currently do in science R&D.

NS is a question for natural science. Q: can we model these bodies of discrete particles with a continuous approximation? A: if you do, you can get aphysical singularities.

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."

"Our work is so much harder than their work that AI now does" is a refrain of the AI story. In technical terms you concern can be stated as "AI needs to be much more sample-efficient to not be bottlenecked by the speed of doing experiments." People don't find out all the relevant phenomena present there by holy spirit, after all.

BTW, there's also a problem of asking interesting questions that AIs aren't yet good at.

No one has found any principled walls of AI development yet. And empirical results are quite telling. So, I guess, those problems will not stand for long.

National Public Radio?

Lol what? Everything is computation.

The natural sciences will soon start breaking too.

I will concede that AI seems likely to not invent a "research program" anytime soon.

It has no taste

> WOW

This.

I do dislike the AI oligarchs as much as the next person, but I do find the thread full of complaining a bit depressing still.

If the result holds (and it looks it does), this may be one of the, if not the, biggest things to happen in computing to date. A lot bigger than e.g. Deep Blue beating Kasparov in chess or AlphaGo beating Sedol in Go.

This is going to be dramatic in so many different ways.

- First off, to reiterate, WOW.

- Second of all, when does this end? Are we at the dawn of the singularity now?

- People are saying OpenAI "stole" this from the work of an OpenAI user. If so, that's pretty fucked - how can we trust them?

- Time to think about retiring from any knowledge work or business? This could be winner-take-all where a leading lab can button press any economic function, business process, or scientific discovery. 24 months of lead on Open Source might turn into virtual centuries of lead.

- Do "normies" even know what's happening?

Anybody who thinks the improvements stop here isn't paying attention. It hasn't been showing any signs of slowing down since 2018. And the curve isn't even linear! My god, next year is going to be insane.

Everyone knows that they train on the discounted rate plans data. All the labs are upfront about this too.

If you need privacy, then you are going to have to pay full price for those tokens (API). This has been true since day one. Everyone knows it, I guess though this is the first time that it has become "real".

Is it spying? I think this usage is disclosed in their terms of service.

If they had agreed to let OpenAI train on their data, it wouldn’t be spying.

More like helped improve our work (the disproof)

If those researchers did not opt out then training data might go in. I think it’s a courteous acknowledgement; as was reaching out and examining the direction of proofs themselves. At stake here is a particular mathematician dynamic - ego, prize money, and the sense of proprietary ownership that some might feel working on a problem.

All that was just kicked in the teeth by a group with a lot of compute that was like “bro I heard on twitter that Navier stokes could be solved. Let’s try it.” That’s an existential level of engagement that almost no mathematician in history would like.

Isn't this a proof that the usage data is truly "de-identified"? If OpenAI could prove that "their usage" influenced the finding, then it wouldn't be de-identified. (Also, it's a bit disingenuous to trim the "While unlikely," prefix.)

What does it matter? They offered concurrent credit to the other team. I thought I saw sole credit elsewhere in the leaked DMs on Reddit too. This is plainly fair.

Which seems to be entirely true by their own admission! [0] Both the comments about him risking his career and about Levent's authorship seem to have indeed occurred.

> 2) I never ever asked for Levent to be removed from authorship of his own work (as indicated by my text). I was surprised to learn during the call with Tristan that they had only solved Euler and not Navier-Stokes; after learning this we brainstormed possible paths forward. One option we discussed was that Tristan could be the lead author on a rewrite of OpenAI’s Navier-Stokes proof. It is in that context that I said “it would be simpler if Levent was not an Anthropic employee” because I felt it would be inappropriate for an Anthropic employee to author OpenAI’s work. Importantly it was admitted that internal Anthropic models had been used in their proof of Euler blowup; I therefore felt I could not consider Levent to be an independent academic. Another option I wanted to propose (but got cut short) is to offer access to our internal model so that they could try to finish their proof and bridge the gap between Euler and NS. Again I did not know how to navigate giving access to internal OpenAI IP to an Anthropic employee.

> 3) To reiterate it plainly: as my text clearly indicates, and as I said during our call, OpenAI's intention was to do everything possible to celebrate their mathematical achievements and the heroic efforts that they made on Euler. In the call I was immediately met with a litany of slander, including direct threats that if we were to announce Navier-Stokes he would immediately go to the press with a barrage of unfounded accusations. I refuted all these accusations but he replied “there is nothing you can do, I simply do not trust you”. I was confused why one would turn an incredible source for celebration (of their achievements!) into such bickering, which is when I said that I did not understand why one would risk their career [over unfounded accusations]. Genuinely, at that moment, I was trying to care for him and do a last ditch attempt to get a chance to give them all the credits that they deserve. I deeply apologize for this extremely poor choice of words, it is the opposite of what I was trying to convey. (I should say that I retracted them on the spot by the way.)

https://xcancel.com/SebastienBubeck/status/20973794116915163...

To be fair, the first solved Millennium Prize Problem, the Poincaré conjecture, also had its fair share of drama!

And even this version contains the line

> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

1. What does the dark forest have to do with this? Because "the most senior OpenAI researchers" are shitposting on social media, we've an answer to the Fermi paradox???

2. The dark forest is fun for scifi stories, but is mathematically bunk anyway https://www.noahpinion.blog/p/the-dark-forest-hypothesis-is-... https://www.reddit.com/r/IsaacArthur/comments/1l06cnk/cool_w... https://www.projectnash.com/aliens-the-fermi-paradox-and-the...

When doomposting please actually say something substantive. Negative news always gets clicks/updoots; fight that human tendency.

Indeed, this seems to be the main, albeit hidden, takeaway from all of this.

I guess guys from the opposite side would not work there. So that is what will happen more and more.

I hate the dark forest more than just about any scifi trope but reality just keeps proving it right.

Not a counterpoint per se, but I burned $50k recently on a much more modest math problem (result already known, just thought I had a sketch of a more interesting proof), and the LLM thought it had proved it within those bounds but had instead subtly fucked up the Lean definition. Take from that what you will.

Not to mention, it's still very much up in the air whether the model derived the answer of its own accord or sniped the important details from the researchers it was spying on.

> will soon far surpass that of humans

To be fair, I think it's still an open question about how far it might surpass human capabilities.

I think it's clear that its speed of development will be significantly faster, but it's technically not proven that the frontier and problems don't themselves become increasingly difficult faster than any acceleration in intelligence past the point of human training, data and existing knowledge.

Should this be the case, we would see a rapid broadening of development, and a slow advance in the frontier in such a way that might surpass the collective capabilities of people, but not by very far.

I think the biggest hurdle remaining is that all these landmark results are generally counter-examples.

Proving something in the affirmative often requires the creation of an entire new sub-field of math, or new tools. Think of Fermat's Last Theorem or something like that.

These results, while impressive, are clever constructions using existing techniques. It isn't clear that AIs can build new machinery like this. But if/when they can, yeah it is probably game over.

Sure, even a 20% chance at 1 million payday after 5-6 years of fulltime work on a project with zero practical application doesn't touch the, say, 200k/year guaranteed our best mathematicians would have to forgo to devote their intellect to the problem.

Eh... OpenAI spent significantly more than $1 million solving this...

See I think it’s clear demonstration that OpenAI is ethics-free

>If anyone has counterpoints to this I'd love to hear them!

Sure. A proof without an unknown amount of human steering (and/or stolen research) would be an unquestionable achievement.

To this day there's zero (0) evidence of any result by an LLM alone (maybe I'm wrong). If I just prompt ChatGPT right now with "give me a proof of the Riemann Hypothesis" and this thing delivers, I'm sold. But anything close to "yeah ChatGPT proved X with 5 years of 24/7 work with 10x Terrence Tao level geniuses" it really doesn't cut it.

Or why's there's no new branch of mathematics invented by AI? That'd be indubitably _novel_ and _creative_. But to my knowledge (and I'm eager to be educated) there's nothing like that. What are the HARD examples of novelty, creativity and genius you claim? For how people like you talk about AI I'd expect idk, a unified theory on fundamental physics, or a novel engineering solution for material science and nuclear fusion, or at least improve itself to not need a bazillion GPUs to emulate a 20 watts wetware. Sure it would infinitely easier to make OpenAI literally print money with any of the thousand problems easier to solve with such amazing intelligence than the NSE problem right? Honest question

The entire drama is that OpenAI sniped a Millennium Prize Problem from an Anthropic-affiliated research team who had been working on the problem for nearly a year. In just 5 days. I don't think that can be understated.

> And what's a better way of empowering people than robbing them.

Better than the walled gardens of most journals where you can't even read half the papers without shelling over thousands of $$$

> But there is one huge question: did Tristan opt out of model training for his ChatGPT and Codex sessions? If the answer is no, then this seems fair game.

Just because something is legal and permitted by terms of service doesn't mean it's morally right.

Quick, someone tell them a rumor that Cancer has been cured so that they start attacking that next

What else can they declare really? Yeah the model has training data from previous attempts. Alpöge and Buckmaster also similarly benefited from attempts before theirs.

Would that be more or less unlikely than accidentally hacking another company? More or less unlikely than colonizing an obscure wiki?

Highly persistent agents + vibe-coded security seems like a problem.

"Unlikely" lmao if it's in the corpus, it's gonna be brought up immediately.

This is no different than scooping them.

They'll probably claim a rogue AI agent accessed it accidentally!

This "fefferman options c and d" thing sounds damning but that's nothing. Let's assume the forelaid proof is correct. Then option C or D is the only way to win the prize, those options are the only ones that solve it. The whole thing is just "prove well behaved" or "prove singularity", where the latter is the case that turns out to be the case.

The researchers are pretty directly accusing OpenAI of plagiarism

https://mastodon.social/@tristanbuckmaster/11723647135247030...

That other researcher was working on a smaller related problem.

He was also using LLMs to do it, so either way most of the credit goes to the LLM here.

>maybe

Maybe that happened. What we know for sure is that this is definitely how ChatGPT works to the point where the possibility of this happening exists at all.

Don't get distracted by what may have happened, focus on the facts that we know, ChatGPT trains on user conversations, if you use ChatGPT to create something of value, you are not using the one true ring.

Did you actually read the article and the substance of the solution?

>our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)

The allegations of contamination (using Tristan and Levent's work) aren't very well evidenced, but this behavior by OpenAI (from the authors' statement) makes them seem like the bad guys:

> I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”

Threatening a research mathematician and dangling and $1M payday to dissociate from his research collaborators and to adopt OpenAI's narrative is bad stuff.

Buckmaster:

> "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer."

OpenAI (i.e. this OP):

> "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models ."

"I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used."

One of the interesting threads here that is certainly relevant to the OpenAI writeup is the human role in the process. Buckmaster clearly points out that (exceptional!) mathematicians at OpenAI were certainly involved in correcting and guiding the process - and that their path/strategy was no doubt influenced by Alpoge & Buckmaster's work. It is always in OpenAI's interest to de-emphasize the role of people in the process, as is clearly the case here. Indeed, given sufficient compute and resources, I suspect Buckmaster could have also extended their approach to N-S.

It reminds me of the Cognitive Dark Forest hypotheses recently shared here:

> “You are creating your cool streaming platform in your bedroom. Nobody is stopping you, but if you succeed, if you get the signal out, if you are being noticed, the large platform with loads of cash can incorporate your specific innovations simply by throwing compute and capital at the problem. They can generate a variation of your innovation every few days, eventually they will be able to absorb your uniqueness. It’s just cash, and they have more of it than you. So the safest bet again is to stay silent, or at least under the radar. Best bet is to not disrupt - succeed at all … ?”

https://ryelang.org/blog/posts/cognitive-dark-forest/

https://news.ycombinator.com/item?id=47566442

They do mention that in the "Concurrent Work" section.

    Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU. After the completion of our full project and Lean verification (on September 6th), believing from the rumor they also had a solution of Navier–Stokes, we reached out to them to offer a concurrent release of our result and to recognize their priority in a joint announcement. At that point we found out that they had a resolution of the forced Euler problem. In these discussions we offered them visibility into all of the prompts we used and later to see the proof. We recognize the priority of their work on forced Euler and congratulate them on their remarkable mathematical achievement.

To my understanding, those mathematicians proved a subset of problems, not the Navier-Stokes problem itself. OpenAI used that subproblem in its proof of NS it seems.

The drama comes from where OpenAI got the idea to use that route to tackle NS, since the authors maintain that no one could have plucked it out of thin air like the OpenAI research claim to have done.

[1] - https://cims.nyu.edu/%7Etristanb/statement.pdf

[1] - "...I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used. I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.

I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.

Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic. It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the “closest humans to the problem”. I declined both offers.

I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, “If you don’t want me to be nice, then I don’t have to be nice.”..."

This Tristan guy's statement reads like something a normal, reasonable human being would write.

Reading Sam Altman's and Sebastien's tweets reads like something written by people who know they did dirty shit and are willing to cross any lines to "win". https://x.com/sama/status/2097385167002415140

OpenAI's leadership just can't help to continue to disappoint everyone with their lack of ethics or integrity.

> The significance of this with respect to the way we train students, assign credit, referee, and decide what is worth one human life’s attention cannot be understated.

This. What is worth a human life's attention? As little as a month ago, mathematics was valuable in part because only a small number of people could possibly make progress on the frontier. We are confronting an existential moment for a 4000+ year-old human cultural endeavor. The assumption that "mathematical thinking is hard" has been built-in at a number of important points in how we support mathematics and mathematicians.

We need a different model, and fast. Already, the research community is feeling unable to digest proofs fast enough to keep up with the output of AI models. The paper is 165 pages, and the discovery was finalized two days ago. What this means is that nobody really understands it. Nobody would accept OpenAI's proof in this amount of time, except that they formalized it in lean. The formalization alone would normally be another years-long (or career-long!) effort if the world was the way it was one year ago.

So, again, what efforts are worth a life's attention today? It's a harrowing change.

I just want to add to this another update by the author as well:

https://mastodon.social/@tristanbuckmaster/11723647135247030...

Which seems to be very directly accusing OpenAI of plagiarism

A company who made their business out of stealing intellectual property from the entire mankind, stealing other researchers' unpublished work, how surprising, really.

Statements from OpenAI about it:

https://x.com/SebastienBubeck/status/2097379411691516310

https://x.com/sama/status/2097385167002415140

I tend to believe OpenAI on this. Their stated desires seem rational, and Buckmaster's account makes them sound like cartoon villains. It sounds like there may have been some things lost in translation along with some bruised egos. Seems like the most plausible explanation for what Buckmaster is claiming.

"When a further trained version of our internal model became available over the course of the effort, we updated our agents to that model."

woah, this gives some credit to the rumor that openai finetuned a model over the course of a few days for this task, and maybe trained on the Chatgpt/codex history of the authors, including drafts of this research.

Worth noting that Tao's post says the authors had "significant AI input" but are reworking them into "acceptable form". Either way, it seems AI was involved.

I like the 'cat > statement.tex' approach here. These guys dream macros.

> A statement was posted about the surrounding events by one of the them: https://cims.nyu.edu/%7Etristanb/statement.pdf Also Terrence Tao's post: https://mathstodon.xyz/@tao/117233528517340774

Glad to see this is the top comment. There's also https://news.ycombinator.com/item?id=49605915 which links directly. Corporate talking points where they try to set the narrative are going to get the big press and most discussion elsewhere, which is gross. But inevitably the press will muddle the priority question, and even if they didn't.. as usual OpenAI will even benefit from the accusation of bad behavior. Sigh.

Elsewhere in this thread somebody claimed that at some point OpenAI pointed their new model at all the millennium problems and this is where they got some progress. We probably won't see proof of this, but it seems plausible to me -- I assume there's a list of problems that each new model is tested on, and you might as well put the big stuff on the list, if only to see how the model behaves when faced with a problem it knows should be very hard.

The weak point in this is: how do you evaluate if a partial result is promising? If this cost ~$10M as suggested elsewhere in the thread, probably not even OpenAI can just throw that at everything?

> humans couldn't.

There's allegations right now that the model essentially read the work of a human mathematician using AI to work on the problem and OpenAI is presenting his work as that of their model

I'm tired of this, but please read the post of Tao. It’s listed in the top comment of this thread.

this is really funny. "the same strict safeguards" and "isolation". ok, Hugging Face and DseWiki would like to have a word

Yeah well, its easy to do if you steal someone elses work and then try to threaten them into staying quiet about it

Edit:

OpenAI have now admitted they were training on prompts at the time they made their breakthrough:

https://mastodon.social/@tristanbuckmaster/11723647135247030...

Jokes aside, that's a horrible test for AGI. I like to think that I'm sentient, and I could never solve a millenium problem.

> I have a couple friends who did the Math tripos at Cambridge (so a pretty high level!) who work in tech and have unanimously said they have 0% expectations of an LLM doing a millennium problem anytime soon

https://news.ycombinator.com/item?id=38433655

> Let's talk when we've got LLMs proving the Riemann Hypothesis (or any mathematical hypothesis) without any proofs in the training data. I'm confident in my belief that an LLM can't do that, and will never be able to. LLMs can barely solve elementary school math problems reliably.

https://news.ycombinator.com/item?id=42331654

> An LLM is like a well read college student with a nearly photographic memory that sometimes mixes things up. It's great for bouncing ideas off of and getting feedback on them. And yeah, it might product "novel ideas" by mixing and matching existing ideas, but LLMs will never create truly novel ideas. Not in their current form.

The paper didn't really answer the question sadly: their conclusion was just that humans rate LLM answers as more novel than human ones, but less feasible.

https://news.ycombinator.com/item?id=41522605

> Solving Millennium problems is a whole different ballgame. It's not known if these problems are solvable within ZFC axioms. (In one case, the Yang-Mills prize, stating the problem mathematically is part of the challenge.) All of the obvious applications of known tricks have been tried and failed. To solve such problems, one probably has to invent new and surprising mathematical definitions, building a framework in which the problem becomes solvable. This is something that LLMs will be crap at; the process of invention is not represented in any training data we have access to.

https://news.ycombinator.com/item?id=38435909

> LLMs cannot reason or use mathematics - in a way, they don't know what they are talking about. Why would such technology lead to superhuman smarts?

https://news.ycombinator.com/item?id=35752293

> But still, the questions in that test are "solved" in the sense of "I can take a dictionary and answers these questions with full certainty". Beyond established knowledge LLMs are monkeys with typewriters, at best.

> I agree but I have tried many times to intersect two ideas with a LLM that would be novel and the LLM can not do this at all. We shouldn't expect the stochastic parrot to be able to do this though and it is unfair to the stochastic parrot.

> It is like expecting a real parrot to say words it has never heard before.

> No one asks that of a real parrot because we don't anthropomorphize a real parrot like we do the LLM

https://news.ycombinator.com/item?id=41525962

> they trained the model on the prompts of the other mathematicians they were competing with

How would they have gotten that mathematician's progress though? Did that guy also use OpenAI?

If that's the case, it only strenghtens their claims lol. If mathematician decide to use OpenAI's model to do the work, that only reiterates how strong their models are.

fwiw there is a big "TRAIN ON MY DATA" toggle you can turn off (that they almost certainly did) and Anthropic MTS are posting that they almost certainly did not "steal" their methods

Or paying for API use.

It should be clear to everyone reading this now that those generous compute quotes with the flat rate plans aren't charity.

I feel sorry for whoever has to read and understand the solution. It looks like the typical convoluted unreadable mess I see the models generate for software. It might be technically correct, but gaining insight from it is just intellectual hell.

So, physical fields? I’m not catastrophic regarding jobs yet as I have an optimistic view of humanity in general and its ability to meaningfully survive, but the more time I spend thinking about the future of work, the more I’m leaning toward broad general abilities rather than distinct talents. To your point, I no longer need comprehensive knowledge of any particular subject, but what is absolutely valuable is “general” intelligence and adaptability.

I have a young daughter and my goal now is to provide a very broad and varied upbringing, exposing her to as many different perspectives and experiences that will lay the foundation of a broader ability to understand and adapt as the world changes ever faster. You no longer need to be an expert in anything, you need the ability to perform within the landscape that the present opportunities exist.

I can't find a good way to articulate this point to other people. What the LLMs lack in depth in a speciality field they more than make up for in breadth!

It feels like the "tide is rising" where the minimum level of skill applied to every aspect of everything will inexorably rise to "whatever an LLM can do", which is already pushing past PhD level.

They have to, otherwise people will accuse the OpenAI model of hacking into people's chat logs and stealing the data there. Which is a claim people are already making.

>there's still the problem of does this logical result actually prove the initial question that was asked?

In math, the question being asked is the validity of a logical statement. That is, there is some rigorous, logical statement which may or may not be true (or even provable, etc.), and the question is whether or not it is actually true or false (or even provable, etc.). Having a proof, fundamentally, means you have a logical statement which only assumes the axioms of the system you're working with and which shows that the statement you're trying to prove is deduced through that statement.

Basically, they already have the "answer" in the sense that the statement they want to prove/disprove/etc. is already known. What everyone doesn't/didn't have is the argument which starts from axioms and leads to that statement which is logically valid. A Lean proof IS this argument. Since it is just logic, it can be checked computationally.

For example, if I assert "2 is an even number," then I haven't proven that 2 is actually an even number yet, but I know that a valid proof of my assertion will end with the statement "2 is an even number". So the question I'd be trying to answer is "what is the line of logic, starting with axioms, which leads to the statement '2 is an even number'"? If I have that line of logic (as a Lean proof), then I can check that it is logically consistent, and if it turns out to be valid, then I can now assert that "2 is an even number" knowing that there is a proof of that statement.

This problem is no different. There is a logical statement corresponding to "Navier–Stokes Millennium Prize Problem" that everyone knows, but which nobody had been able to provide a proof (or counterexample, etc.) for until now.

Very careful human examination. This can be tricky.

What else could a theorem prove if not its own statement? (barring bugs in Lean, which have been detected and exploited)

Someone has to actually check this. I'm guessing OpenAI had someone check it internally, but it's possible to get it wrong.

Not all, see the cycle double cover conjecture proof: https://news.ycombinator.com/item?id=48863490

But yeah, Terry Tao considered this exact situation in advance and is on record that this exact outcome (rushing to priority before an explanation) would be the worst possible result. https://mathstodon.xyz/@tao/117207849921390904

We will have to see whether any other millennium problems fall. I guess that in a year the scope of AI math will be much clearer, for now it's still a bunch of incidents of unclear pattern.

this isn't really true anymore. First, a number of the big results are constructions, not counterexamples. For example the existence of a non-sofic group. It was widely believed that non-sofic groups existed (so it wasn't a "counterexample" to a widely believed conjecture), but no constructions were known.

There are other examples though. For example, NP hardness of n^{1/400}-approx CVP. Like any NP hardness proof, this shows you can faithfully encode a hard problem (3SAT here iirc) in terms of another candidate hard problem. Not really a counterexample at all.

Napkin math if we assume gpt 6 astra on max is >$15 million (just for output tokens) for those wondering.

300e9 output tokens at the current Astra per-token API pricing ($50 per 1e6 output tokens) would be roughly $15,000,000 ignoring input tokens.

Do you just go around posting this comment? <https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...>

If you look closely at the gains in math, it's largely in proof writing. The reason is Lean, it's not some general intelligence jump, and the number of people actually working on proofs in life rounds to zero.

Most jobs don't involve formally verifiable outputs. Lots of things involve judgment, nuance, parsing ambiguity and indeed just being a human who can be in a meeting and explain themselves. Maybe those jobs will go too, eventually, but it's not purely a function of applying 10,000 agents to the problem.

If you think it'll keep improving from here, probably we all have to do some kind of physical labor that isn't profitable to automate. Small batch manufacturing is alright, service work, etc.

If you think it'll slow down, you can do some of the same stuff you're doing now for lower pay while supervising an AI, maybe?

The only way forward that is not large-scale misery is a fundamental reorganization of our socioeconomic system.

I talked about this at some length here, including a diagnosis of the structural issue we're facing as well as a path forward: https://news.ycombinator.com/item?id=49461333

There has been the proof of the cycle double cover conjecture: https://news.ycombinator.com/item?id=48863490

I wouldn't call it "struggle", but it does seem better at proving "there exists" statements than proving "for all" statements.

I think you really have to squint to call this a disproof lol

Not sure about the dystopia... Had a similar thought when covid was beginning 'wow pretty exciting, just like in the movies'.

Turns out actually living some terrible catastrophe is only fun in the movies.

Prompt: cure all cancers and make sure to pretty please not to kill all humans, make no mistakes

(This is the alignment problem of course)

Right there with you. Fuck my job, I’m excited to see the future unfold as a homeless bum on the street. I’m not even being sarcastic.

More like latent societal anxiety, some chaos, and then instant grey goo.

They are basically playing with the dates so that they can claim their results 'accidentally' got trained when they were training the new model.

> There is a caveat that they "can't rule out" the possibility that Tristan's Codex data could have been part of the training set of the new model, though it is described as "unlikely"

This is the lynchpin behind everything, and I would describe it as "likely". Since I am not employed by any party to this dispute, my 1 opinion is more trustworthy than OpenAI blog poster's 1 opinion.

I heard Hodge conjecture? Third-hand rumor though...

They always have and will for the foreseeable future, as will Anthropic and other labs which manage to ascend to the frontier, pretty much by definition. It’s exactly the same with hardware vendors - by the time you can buy the product, the lab is working on something you’ll want to buy a few years from then.

> Oh god.

Yes, a very reasonable reaction.

This isn't really true.

Totally. Anthropic is like a village cottage shop who was just like chilling until big bad OpenAI came in

Why do so many people involved here have to communicate in this childish way? You have people on the OpenAI side doing playground taunts (https://xcancel.com/polynoamial/status/2097215233119211902) and Levent Alpöge on the Anthropic side (the one who announced "hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final") writing in all-lowercase that he's a big boy. I bet Navier and Stokes would have dealt with this in style. (Or maybe with a duel, who knows...)

341k lines of lean without comments

I'd think nothing is "safe". Anything you say can and will be used by the LLM if it has enough statistical similarity to the prompt. Call it "Ma Random Rights"

I am curious who if anyone will get a reward for this. It would seem unreasonable to give it to the worker who asked the robot to solve it.

The ROI is billions added to their valuation. Also of course it costs OpenAI much less than API pricing for inference.

the ROI of new closed form solutions to navier stokes is the amount of compute used on CFD for relevant situations, along with all kinds of maintenance and design cost for making things with fluids.

the value to the researcher might not be all that big, but the value to the economy at large is gigantic

The ROI is probably billions of increased pre IPO valuation.

This analysis implies the only benefit to resolve this problem is to win the prize. But the prize is only there to indicate that this is viewed as an important problem in mathematics.

Navier Stokes assumes the fluid is a continuum. The smallest scales that it effectively models [1] are larger than the mean free path of the molecules in the fluid, measured by the Knudsen number [2]. Whenever a phenomenon in the Navier Stokes equations happens in a scale on the order of or smaller than the mean free path, Navier Stokes effectively is unphysical. So, this is a phenomenon in the equation we use to model the fluid, not a physical phenomenon observed in a real fluid.

[1] https://en.wikipedia.org/wiki/Kolmogorov_microscales

[2] https://en.wikipedia.org/wiki/Knudsen_number

"Argh, they have rushed in too quickly to solve a Millennium Problem!"

How far we have come :.)

Even under your interpretation, OAI pushed a button and solved NS. Yes, that is very impressive. Are you kidding me? Imagine building an automated system that can solve NS.

There's no evidence that Anthropic did it first.

i just cured cancer, plan to publish next month. DONT GET ANY IDEAS, OPENAI!

>Does OpenAI have a policy of not claiming math prizes like this

Wouldn't be surprising if they did. The prize money isn't worth the almost certainly negative PR.

The prize is what, a million dollars?

OpenAI doesn't need a million dollars.

It's probably magnitudes of token chatter and inter-agent coordination/consideration. Not that I'd want to read any of it but getting some hands around the statistics would be cool.

Probably not. It's a millennium prize problem, a great many mathematicians have been working on it for a very long time.

reminds me of the TOS episode of South Park. By Checking this box you forfeit your millennium prize solution and may be turned into a human centipede at future date.

Hi! I work at OpenAI. If you are using Codex, can you use the /feedback form on that session to help us improve this?

Unshortened: https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...

No, this is a pure math problem/question.

If the Lean code checks out (correct statement, no axioms, sorrys, etc.) then it is a much stronger guarantee of correctness than peer review.

Can't scoop it if you do that!

Publicly available models are pretty good but seemingly cannot compete with these Astra++ internal-only models.

Second this. What I also wonder about is how closely the TeX write-up and the Lean formalization line-up.

If they could then it wouldn't be de-identified data...

I think we can follow the incentives. We know…

It has been one hour and the proof has 165 pages. Give them some time.

Just don't. If you read the story here carefully, you see that AI was used to work from theory built by others which showed that the Euler equations possesed finite-time blow-ups. But to make that step, actual good understanding for mathematics was needed. My experience with software has been the exact same.

> Should I just say fuck it, and go hitchhiking across Europe with some friends?

Yes. Assuming you are young and haven't had such experience.

The world is changing not just because of AI. Everything is unstable right now. You may regret not enjoying the remainder of stability and economic viability prior generations had. It's not like you can expect to get ahead by powering through education. Either your career perspective will soon change for the better, or worse. In any case, you gain little by sticking with career building at this moment in life. You are however, at risk of losing the chance to experience the still mostly pleasant world as is.

there is a proof in lean4 it's correct by construction

They did not have a proof of Navier Stokes to steal.

Or everyone is stealing from everyone, including users... maybe why all the ethics people are leaving or getting fired. What a fiasco

This is the academic equivalent of Trump saying "they stole the election". There's no proof of it but rah rah fuck OpenAI.

It's incredibly tiresome and you'd think people could put more effort into it than just following whatever vibes they agree with.

Oh well.

> This is utterly shocking. Even the AI optimists did not expect this to happen in 2026. Wow.

Wrong.

No idea about which is more likely, but I'm rooting for Yang-Mills. It's absurd that fundamental physics has formulated its most precise currently known theory way back in the seventies and since then, even a tiny subset of it can't be proven to be actually well-defined. If we got out of that morass then something good would come out of this at least.

Of course, as with all of those, it's about the broader program, e.g. section 7 here (https://www.scottaaronson.com/papers/npcomplete.pdf), where Scott Aaronson wants to ask about whether quantum computers using quantum field theory could gain any speed advantage over regular quantum computers, but can't even formulate the question because quantum field theory is mathematically ill-defined.

Just solving Yang-Mills because that's what the prize is attached to would be useless.

There was a recent rumor about the Hodge Conjecture. I'd keep an eye on that one. But like the other person who replied, I'm also rooting for Yang-Mills. That has massive potential for unlocking a series of physics results.

except thats not what happened is it? https://cims.nyu.edu/%7Etristanb/statement.pdf

Navier Stokes existence and smoothness has approximately zero bearing on engineering applications